Subgroups — #72226371

Order 2048.

At least 6833 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

The subgroup lattice has 180 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225897	C₂	1	1	yes	maximal, characteristic, complemented
2	#72219296	C₂	1	1	yes	maximal, characteristic, complemented
3	#72219294	C₂	1	1	yes	maximal, characteristic, complemented
4	#72218846	C₂	1	1	yes	maximal, characteristic, complemented
5	#72218795	C₂	1	1	yes	maximal, characteristic
6	#72218451	C₂	1	1	yes	maximal, characteristic, complemented
7	#72215179	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×#120	C₂²	1	1	yes	characteristic
9	C₂²×#2403	C₂²	1	1	yes	characteristic
10	C₂²×#1781	C₂²	1	1	yes	characteristic
11	C₂×#15947	C₄	1	2	yes	complemented
12	#517411	C₂²	1	1	yes	characteristic
13	#392113	C₂²	1	1	yes	characteristic
14	#392109	C₂²	1	1	yes	characteristic
15	#154670	C₄	1	2	yes	complemented
16	#154094	C₂²	1	1	yes	characteristic
17	C₂⁸	D₄	1	1	yes	characteristic
18	C₂⁵×D₄	D₄	1	1	yes	characteristic
19	C₂⁵×D₄	C₂³	1	1	yes	characteristic
20	C₂³×#120	C₂×C₄	1	2	yes	—
21	C₂³×#120	C₂×C₄	1	1	yes	characteristic
22	C₂³×#120	D₄	1	2	yes	—
23	C₂³×#120	D₄	1	1	yes	characteristic
24	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic
25	C₂³×#99	D₄	1	2	yes	complemented
26	C₂×#2308	D₄	1	2	yes	complemented
27	C₂²×#386	C₂×C₄	1	1	yes	characteristic
28	C₂²×#386	D₄	1	1	yes	characteristic
29	C₂²×#379	D₄	1	1	yes	characteristic
30	C₂×#1783	D₄	1	2	yes	complemented
31	C₂×#1781	D₄	1	2	yes	complemented
32	C₂×#1438	C₂×C₄	1	2	yes	—
33	C₂×#1438	D₄	1	2	yes	—
34	C₂×C₂²⋊#95	D₄	1	2	yes	—
35	C₂⁷	C₂²⋊C₄	1	2	yes	—
36	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic, complemented
37	C₂⁷	C₂×D₄	1	1	yes	characteristic
38	C₂⁷	C₂×D₄	1	1	yes	characteristic
39	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
40	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
41	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
42	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
43	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
44	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
45	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
46	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
47	C₂⁴×D₄	#41	1	1	yes	characteristic
48	C₂⁴×D₄	#41	1	1	yes	characteristic
49	C₂⁴×D₄	#41	1	1	yes	characteristic
50	C₂⁴×D₄	#41	1	1	yes	characteristic
51	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
52	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
54	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
55	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
56	C₂²×#95	C₂²⋊C₄	1	4	yes	complemented
57	C₂⁶	#99	1	4	yes	complemented
58	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂⁶	#120	1	1	yes	characteristic
61	C₂⁶	#120	1	1	yes	characteristic
62	C₂⁶	#120	1	1	yes	characteristic
63	C₂⁶	#120	1	1	yes	characteristic
64	C₂⁶	#120	1	1	yes	characteristic
65	C₂⁶	#121	1	1	yes	characteristic
66	C₂⁶	#121	1	1	yes	characteristic
67	C₂⁶	#121	1	1	yes	characteristic
68	C₂⁶	#121	1	1	yes	characteristic
69	C₂⁶	#123	1	1	yes	characteristic
70	C₂⁶	#123	1	1	yes	characteristic
71	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
72	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
73	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
74	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
75	C₂³×D₄	#120	1	1	yes	characteristic
76	C₂³×D₄	#120	1	1	yes	characteristic
77	C₂³×D₄	#121	1	1	yes	characteristic
78	C₂³×D₄	#121	1	1	yes	characteristic
79	C₂³×D₄	#121	1	1	yes	characteristic
80	C₂³×D₄	#121	1	1	yes	characteristic
81	C₂³×D₄	#121	1	1	yes	characteristic
82	C₂³×D₄	#121	1	1	yes	characteristic
83	C₂³×D₄	#123	1	1	yes	characteristic
84	C₂³×D₄	#123	1	1	yes	characteristic
85	C₂³×D₄	#124	1	1	yes	characteristic
86	C₂³×D₄	#127	1	1	yes	characteristic
87	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂⁴×C₄	#120	1	1	yes	characteristic
89	C₂⁴×C₄	#121	1	1	yes	characteristic
90	C₂⁴×C₄	#121	1	1	yes	characteristic
91	C₂⁴×C₄	#124	1	1	yes	characteristic
92	C₂⁴×C₄	#127	1	1	yes	characteristic
93	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
94	C₂⁵	#379	1	1	yes	characteristic
95	C₂⁵	#386	1	1	yes	characteristic
96	C₂⁵	#386	1	1	yes	characteristic
97	C₂⁵	#386	1	1	yes	characteristic
98	C₂⁵	#386	1	1	yes	characteristic
99	C₂⁵	#392	1	1	yes	characteristic
100	C₂⁵	#392	1	1	yes	characteristic
101	C₂⁵	#392	1	1	yes	characteristic
102	C₂⁵	#394	1	1	yes	characteristic
103	C₂⁵	#394	1	1	yes	characteristic
104	C₂⁵	#394	1	1	yes	characteristic
105	C₂⁵	C₂×#99	1	2	yes	—
106	C₂⁵	C₂×#99	1	2	yes	—
107	C₂⁵	B(4,𝔽₂)	1	2	yes	—
108	C₂⁵	B(4,𝔽₂)	1	4	yes	—
109	C₂⁵	B(4,𝔽₂)	1	2	yes	—
110	C₂⁵	B(4,𝔽₂)	1	4	yes	—
111	C₂⁵	#458	1	2	yes	—
112	C₂⁵	#458	1	2	yes	—
113	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
114	C₂³×C₄	#390	1	1	yes	characteristic
115	C₂³×C₄	#390	1	1	yes	characteristic
116	C₂³×C₄	#392	1	1	yes	characteristic
117	C₂³×C₄	#394	1	1	yes	characteristic
118	C₂⁴	#1438	1	1	yes	characteristic
119	C₂⁴	#1781	1	2	yes	—
120	C₂⁴	#1790	1	2	yes	—
121	C₂⁴	#1790	1	2	yes	—
122	C₂⁴	#1889	1	2	yes	—
123	C₂⁴	#1889	1	2	yes	—
124	C₂⁴	#1889	1	2	yes	—
125	C₂⁴	#1898	1	2	yes	—
126	C₂⁴	#1898	1	1	yes	characteristic
127	C₂⁴	#1898	1	1	yes	characteristic
128	C₂⁴	#1913	1	2	yes	—
129	C₂⁴	#2021	1	2	yes	—
130	C₂⁴	#2021	1	2	yes	—
131	C₂⁴	#2021	1	1	yes	characteristic
132	C₂⁴	#2021	1	2	yes	—
133	C₂⁴	#2021	1	1	yes	characteristic
134	C₂⁴	#2021	1	2	yes	—
135	C₂⁴	#2021	1	2	yes	—
136	C₂⁴	#2022	1	1	yes	characteristic
137	C₂⁴	#2022	1	1	yes	characteristic
138	C₂⁴	#2044	1	2	yes	—
139	C₂⁴	#2044	1	2	yes	—
140	C₂⁴	#2196	1	4	yes	—
141	C₂⁴	#2199	1	4	yes	—
142	C₂²×C₄	#2021	1	2	yes	—
143	C₂³	#8534	1	4	yes	—
144	C₂³	#9495	1	2	yes	—
145	C₂³	#9497	1	2	yes	—
146	C₂³	#10336	1	2	yes	—
147	C₂³	#10336	1	2	yes	—
148	C₂³	#10337	1	2	yes	—
149	C₂³	#10337	1	2	yes	—
150	C₂³	#10965	1	1	yes	characteristic
151	C₂³	#10972	1	1	yes	characteristic
152	C₂³	#10991	1	1	yes	characteristic
153	C₂³	#10991	1	1	yes	characteristic
154	C₂³	#10991	1	1	yes	characteristic
155	C₂³	#10995	1	1	yes	characteristic
156	C₂³	#10995	1	1	yes	characteristic
157	C₂³	#12753	1	4	yes	—
158	C₂³	#12759	1	2	yes	—
159	C₂³	#12763	1	2	yes	—
160	C₂³	#13343	1	4	yes	—
161	C₂³	#13419	1	4	yes	—
162	C₂²	#111458	1	1	yes	characteristic
163	C₂²	#111458	1	1	yes	characteristic
164	C₂²	#115133	1	1	yes	characteristic
165	C₂²	#115204	1	1	yes	characteristic
166	C₂²	#115204	1	1	yes	characteristic
167	C₂²	#115348	1	1	yes	characteristic
168	C₂²	#115690	1	1	yes	characteristic
169	C₂²	#133390	1	4	yes	—
170	C₂²	#133394	1	4	yes	—
171	C₂²	#144366	1	4	yes	—
172	C₂²	#147779	1	2	yes	—
173	C₂²	#147815	1	2	yes	—
174	C₂	#72218387	1	1	yes	characteristic
175	C₂	#72218840	1	1	yes	characteristic
176	C₂	#72219162	1	1	yes	characteristic
177	C₂	#72225799	1	2	yes	—
178	C₂	#72226368	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#10995
commutator	C₂⁴×D₄	C₂²×C₄
Frattini, cosocle	C₂⁵×D₄	C₂³
prime radical	#72218795	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72225897	C₂	semidirect
2. #72219296	C₂	semidirect
3. #72219294	C₂	semidirect
4. #72218846	C₂	semidirect
6. #72218451	C₂	semidirect
7. #72215179	C₂	semidirect
11. C₂×#15947	C₄	semidirect
15. #154670	C₄	semidirect
25. C₂³×#99	D₄	semidirect
26. C₂×#2308	D₄	semidirect
30. C₂×#1783	D₄	semidirect
31. C₂×#1781	D₄	semidirect
36. C₂⁷	C₂²⋊C₄	semidirect
51. C₂⁵×C₄	C₂²⋊C₄	semidirect
54. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
55. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
56. C₂²×#95	C₂²⋊C₄	semidirect
57. C₂⁶	#99	semidirect
93. C₂²×C₂²⋊C₄	#99	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#72225897	2
#72219296	2
#72219294	2
#72218846	2
#72218795	2
#72218451	2
#72215179	2